![](bg6.png)
ous state and noise bounds estimation and involve-
ment of other bounded distributions to generalise the
class of used pdfs.
ACKNOWLEDGEMENTS
This research was partially supported by the grant
GA
ˇ
CR 18-15970S.
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A APPENDIX
A.1 Approximation of the Trapezoidal
Pdf
According to (Bernardo, 1979), minimisation od
Kullback-Liebler divergence (Kullback and Leibler,
1951) (KLD) gives, in a Bayesian sense, an optimal
approximation of pdf. KLD of two pdfs, f
1
and f
2
,
equals D( f
1
|| f
2
) =
R
f
1
(x)ln
f
1
(x)
f
2
(x)
dx.
Denote the set describing the support χ
?
≡
supp(χ). Given pdf on a bounded support, f
1
(x) =
g(x)χ
1
(x), where 0 < g(x) < +∞ ∀x ∈ χ
?
1
, we search
for the optimal approximation by a uniform pdf,
f
2
(x) = I
−1
χ
2
(x), where I = vol(χ
?
2
). We look for
ˆ
f
2
= argmin
f
2
D( f
1
|| f
2
). Function arguments are omit-
ted.
Here, D( f
1
|| f
2
) =
R
gln
I gχ
1
χ
2
dx =
R
χ
?
1
gln gdx +
R
χ
?
1
gln
χ
1
χ
2
dx + lnI
R
χ
?
1
gdx. The first term is indepen-
dent of f
2
, the second term is finite (zero) if χ
?
1
⊂ χ
?
2
.
The third term depends on f
2
through I : the larger
support of f
2
, the higher I . Hence, to minimise KLD,
we minimise the measure of χ
?
2
choosing χ
2
= χ
1
, i.e.
ˆ
f
2
be the uniform pdf on the support of f
1
.
A.2 Approximation of the Polytope by
a Parallelotope
The second term in (11) can be understood as `
y
data strips in x
t
-space, y
t;i
− r
i
≤ C
i
x
t
≤ y
t;i
+ r
i
, i =
1,. .. ,`
y
. These strips intersect with `
x
strips given by
the first term, which forms a parallelotope (actually
orthotope). The intersection defines a polytope which
is to be approximated by another parallelotope (12).
Theory and algorithm of the approximation is de-
scribed in (Vicino and Zappa, 1996). Briefly: (i) One
Approximate Recursive Bayesian Estimation of State Space Model with Uniform Noise
393